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\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  -11.0 & -7.00 x_{1} & -9.00 x_{2} & +  9.00 x_{3} & -1.00 x_{4} & +  8.00 x_{5}\\
 x_{7}   &  4.0 & +  3.00 x_{1} & -8.00 x_{2} & +  3.00 x_{3} & -10.00 x_{4} & +  4.00 x_{5}\\
 x_{8}   &  4.0 & + 10.00 x_{1} & -7.00 x_{2} & -7.00 x_{3} & + 10.00 x_{4} & -1.00 x_{5}\\
 x_{9}   &  17.0 & +  5.00 x_{1} & +  2.00 x_{2} & -9.00 x_{3} & +  1.00 x_{4} & +  2.00 x_{5}\\
 x_{10}   &  -5.0 & -3.00 x_{1} &   & -2.00 x_{3} & + 10.00 x_{4} & -5.00 x_{5}\\
 x_{11}   &  -2.0 & -2.00 x_{1} & -4.00 x_{2} & +  6.00 x_{3} & -10.00 x_{4} & +  6.00 x_{5}\\
\hline
z    &  0.0 & -4.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & -1.00 x_{4} & -4.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,11$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  7.00 y_{6} & -3.00 y_{7} & -10.00 y_{8} & -5.00 y_{9} & +  3.00 y_{10} & +  2.00 y_{11}\\
 y_{2}   &  1.0 & +  9.00 y_{6} & +  8.00 y_{7} & +  7.00 y_{8} & -2.00 y_{9} &   & +  4.00 y_{11}\\
 y_{3}   &  1.0 & -9.00 y_{6} & -3.00 y_{7} & +  7.00 y_{8} & +  9.00 y_{9} & +  2.00 y_{10} & -6.00 y_{11}\\
 y_{4}   &  1.0 & +  1.00 y_{6} & + 10.00 y_{7} & -10.00 y_{8} & -1.00 y_{9} & -10.00 y_{10} & + 10.00 y_{11}\\
 y_{5}   &  1.0 & -8.00 y_{6} & -4.00 y_{7} & +  1.00 y_{8} & -2.00 y_{9} & +  5.00 y_{10} & -6.00 y_{11}\\
\hline
z    &  -0 & + 11.00 y_{6} & -4.00 y_{7} & -4.00 y_{8} & -17.00 y_{9} & +  5.00 y_{10} & +  2.00 y_{11}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 6 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  2.8 & -1.25 y_{3} & -1.25 y_{7} & -6.75 y_{8} & +  5.70 y_{9} & -0.55 y_{4} & -3.70 y_{6}\\
 y_{2}   &  2.2 & -1.00 y_{3} & +  7.00 y_{7} & + 12.00 y_{8} & +  6.80 y_{9} & -0.20 y_{4} & +  0.20 y_{6}\\
 y_{11}   &  0.3 & -0.25 y_{3} & -0.25 y_{7} & +  1.25 y_{8} & +  2.20 y_{9} & -0.05 y_{4} & -2.20 y_{6}\\
 y_{10}   &  0.4 & -0.25 y_{3} & +  0.75 y_{7} & +  0.25 y_{8} & +  2.10 y_{9} & -0.15 y_{4} & -2.10 y_{6}\\
 y_{5}   &  1.2 & +  0.25 y_{3} & +  1.25 y_{7} & -5.25 y_{8} & -4.70 y_{9} & -0.45 y_{4} & -5.30 y_{6}\\
\hline
z    &  2.6 & -1.75 y_{3} & -0.75 y_{7} & -0.25 y_{8} & -2.10 y_{9} & -0.85 y_{4} & -3.90 y_{6}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  1.75 & +  1.25 x_{1} & +  1.00 x_{2} & +  0.25 x_{11} & +  0.25 x_{10} & -0.25 x_{5}\\
 x_{7}   &  0.75 & +  1.25 x_{1} & -7.00 x_{2} & +  0.25 x_{11} & -0.75 x_{10} & -1.25 x_{5}\\
 x_{8}   &  0.25 & +  6.75 x_{1} & -12.00 x_{2} & -1.25 x_{11} & -0.25 x_{10} & +  5.25 x_{5}\\
 x_{9}   &  2.1 & -5.70 x_{1} & -6.80 x_{2} & -2.20 x_{11} & -2.10 x_{10} & +  4.70 x_{5}\\
 x_{4}   &  0.85 & +  0.55 x_{1} & +  0.20 x_{2} & +  0.05 x_{11} & +  0.15 x_{10} & +  0.45 x_{5}\\
 x_{6}   &  3.9 & +  3.70 x_{1} & -0.20 x_{2} & +  2.20 x_{11} & +  2.10 x_{10} & +  5.30 x_{5}\\
\hline
z    &  -2.6 & -2.80 x_{1} & -2.20 x_{2} & -0.30 x_{11} & -0.40 x_{10} & -1.20 x_{5}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  1.75 & +  1.25 x_{1} & +  1.00 x_{2} & +  0.25 x_{11} & +  0.25 x_{10} & -0.25 x_{5}\\
 x_{7}   &  0.75 & +  1.25 x_{1} & -7.00 x_{2} & +  0.25 x_{11} & -0.75 x_{10} & -1.25 x_{5}\\
 x_{8}   &  0.25 & +  6.75 x_{1} & -12.00 x_{2} & -1.25 x_{11} & -0.25 x_{10} & +  5.25 x_{5}\\
 x_{9}   &  2.1 & -5.70 x_{1} & -6.80 x_{2} & -2.20 x_{11} & -2.10 x_{10} & +  4.70 x_{5}\\
 x_{4}   &  0.85 & +  0.55 x_{1} & +  0.20 x_{2} & +  0.05 x_{11} & +  0.15 x_{10} & +  0.45 x_{5}\\
 x_{6}   &  3.9 & +  3.70 x_{1} & -0.20 x_{2} & +  2.20 x_{11} & +  2.10 x_{10} & +  5.30 x_{5}\\
\hline
z    &  -9.6 & -10.80 x_{1} & -8.20 x_{2} & -1.30 x_{11} & -1.40 x_{10} & -3.20 x_{5}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  1.75 & +  1.25 x_{1} & +  1.00 x_{2} & +  0.25 x_{11} & +  0.25 x_{10} & -0.25 x_{5}\\
 x_{7}   &  0.75 & +  1.25 x_{1} & -7.00 x_{2} & +  0.25 x_{11} & -0.75 x_{10} & -1.25 x_{5}\\
 x_{8}   &  0.25 & +  6.75 x_{1} & -12.00 x_{2} & -1.25 x_{11} & -0.25 x_{10} & +  5.25 x_{5}\\
 x_{9}   &  2.1 & -5.70 x_{1} & -6.80 x_{2} & -2.20 x_{11} & -2.10 x_{10} & +  4.70 x_{5}\\
 x_{4}   &  0.85 & +  0.55 x_{1} & +  0.20 x_{2} & +  0.05 x_{11} & +  0.15 x_{10} & +  0.45 x_{5}\\
 x_{6}   &  3.9 & +  3.70 x_{1} & -0.20 x_{2} & +  2.20 x_{11} & +  2.10 x_{10} & +  5.30 x_{5}\\
\hline
z    &  -9.6 & -10.80 x_{1} & -8.20 x_{2} & -1.30 x_{11} & -1.40 x_{10} & -3.20 x_{5}\\
\end{array}\]
Final Dictionary
Solution:  -9.6
Num Pivots:  1
\end{document}
